1,2

a(3)=3 because the 3 X 3 square can be divided into sub-squares in 3 different ways: a single 3 X 3 square, a 2 X 2 square plus five 1 X 1 squares, or nine 1 X 1 squares. a(9)=312 because the 9 X 9 square can be divided into 312 different combinations of sub-squares such as three 4 X 4 squares plus thirty-three 1 X 1 squares, etc.

Comment from Jon Schoenfield, Sep 18 2008: There are 11 different ways to divide a 5x5 square into sub-squares:

1. 25(1x1)

2. 1(2x2) + 21(1x1)

3. 2(2x2) + 17(1x1)

4. 3(2x2) + 13(1x1)

5. 4(2x2) + 9(1x1)

6. 1(3x3) + 16(1x1)

7. 1(3x3) + 1(2x2) + 12(1x1)

8. 1(3x3) + 2(2x2) + 8(1x1)

9. 1(3x3) + 3(2x2) + 4(1x1)

10. 1(4x4) + 9(1x1)

11. 1(5x5)

Cf. A129668, A014544.

Adjacent sequences: A140105 A140106 A140107 this_sequence A140109 A140110 A140111

Sequence in context: A130968 A007748 A126617 this_sequence A117733 A066236 A118203

hard,more,nice,nonn,new

Sergio Pimentel (ferdiego(AT)suddenlink.net), Jun 03 2008

Corrected three terms, added three new terms, and corrected and edited example. - Jon E. Schoenfield (jonscho(AT)hiwaay.net), Sep 19 2008